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 The KDE Procedure

## Kernel Density Estimates

A weighted univariate kernel density estimate involves a variable X and a weight variable W. Let (Xi,Wi), i = 1,2, ... ,n denote a sample of X and W of size n. The weighted kernel density estimate of f(x), the density of X, is as follows:

where h is the bandwidth and
is the standard normal density rescaled by the bandwidth. If and , then the optimal bandwidth is
This optimal value is unknown, and so approximations methods are required. For a derivation and discussion of these results, refer to Silverman (1986, Chapter 3) and Jones, Marron, and Sheather (1996).

For the bivariate case, let X = (X,Y) be a bivariate random element taking values in with joint density function , and let Xi = (Xi,Yi), i = 1,2, ... , n be a sample of size n drawn from this distribution. The kernel density estimate of f(x,y) based on this sample is

where , hX>0 and hY>0 are the bandwidths and is the rescaled normal density:
where is the standard normal density function:

Under mild regularity assumptions about f(x,y), the mean integrated squared error of is

as , and .

Now set

which is the asymptotic mean integrated squared error. For fixed n, this has minimum at (hAMISE_X, hAMISE_Y) defined as
and
These are the optimal asymptotic bandwidths in the sense that they minimize MISE. However, as in the univariate case, these expressions contain the second derivatives of the unknown density f being estimated, and so approximations are required. Refer to Wand and Jones (1993) for further details.

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